Solve ab+ax+bx+x^2=0

$$a b + a x + b x + x ^ { 2 } = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=-x\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&b=-x\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract bx from both sides. Anything subtracted from zero gives its negation.

$$ab+ax+x^{2}=-bx$$

Subtract x^{2} from both sides.

$$ab+ax=-bx-x^{2}$$

Reorder the terms.

$$ax+ab=-x^{2}-bx$$

Combine all terms containing a.

$$\left(x+b\right)a=-x^{2}-bx$$

Divide both sides by b+x.

$$\frac{\left(x+b\right)a}{x+b}=-\frac{x\left(x+b\right)}{x+b}$$

Dividing by b+x undoes the multiplication by b+x.

$$a=-\frac{x\left(x+b\right)}{x+b}$$

Divide -x\left(x+b\right) by b+x.

$$a=-x$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=-x\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&a=-x\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax from both sides. Anything subtracted from zero gives its negation.

$$ab+bx+x^{2}=-ax$$

Subtract x^{2} from both sides.

$$ab+bx=-ax-x^{2}$$

Reorder the terms.

$$bx+ab=-x^{2}-ax$$

Combine all terms containing b.

$$\left(x+a\right)b=-x^{2}-ax$$

Divide both sides by a+x.

$$\frac{\left(x+a\right)b}{x+a}=-\frac{x\left(x+a\right)}{x+a}$$

Dividing by a+x undoes the multiplication by a+x.

$$b=-\frac{x\left(x+a\right)}{x+a}$$

Divide -x\left(x+a\right) by a+x.

$$b=-x$$

Solve for a

$$\left\{\begin{matrix}\\a=-x\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&b=-x\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract bx from both sides. Anything subtracted from zero gives its negation.

$$ab+ax+x^{2}=-bx$$

Subtract x^{2} from both sides.

$$ab+ax=-bx-x^{2}$$

Reorder the terms.

$$ax+ab=-x^{2}-bx$$

Combine all terms containing a.

$$\left(x+b\right)a=-x^{2}-bx$$

Divide both sides by b+x.

$$\frac{\left(x+b\right)a}{x+b}=-\frac{x\left(x+b\right)}{x+b}$$

Dividing by b+x undoes the multiplication by b+x.

$$a=-\frac{x\left(x+b\right)}{x+b}$$

Divide -x\left(x+b\right) by b+x.

$$a=-x$$

Solve for b

$$\left\{\begin{matrix}\\b=-x\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&a=-x\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax from both sides. Anything subtracted from zero gives its negation.

$$ab+bx+x^{2}=-ax$$

Subtract x^{2} from both sides.

$$ab+bx=-ax-x^{2}$$

Reorder the terms.

$$bx+ab=-x^{2}-ax$$

Combine all terms containing b.

$$\left(x+a\right)b=-x^{2}-ax$$

Divide both sides by a+x.

$$\frac{\left(x+a\right)b}{x+a}=-\frac{x\left(x+a\right)}{x+a}$$

Dividing by a+x undoes the multiplication by a+x.

$$b=-\frac{x\left(x+a\right)}{x+a}$$

Divide -x\left(x+a\right) by a+x.

$$b=-x$$